Explanation

(d)
The given condition suggest that $a$ lies between the roots.
Let $f(x) = 2x^{2} - 2(2a + 1)x + a(a + 1)$
For $`a'$ to lie between the roots we must have Discriminant $\geq 0\ and\ f(a) < 0$
Now, Discriminant $\geq 0$
$\Rightarrow 4(2a + 1)^{2} - 8a(a + 1) \geq 0$
$\Rightarrow 8\left( \ a^{2} + a + \frac{1}{2} \right)) \geq 0$, which is always true.
Also, $f(a) < 0$
$\Rightarrow \ \ 2a^{2} - 2a(2a + 1) + a(a + 1) < 0$
$\Rightarrow - a^{2} - a < 0 \Rightarrow a^{2} + a > 0 \Rightarrow a(1 + a) > 0$
$\Rightarrow a > 0\ \ or\ \ a < - 1$